Publicação
Nonlocal Lagrange multipliers and transport densities
| Resumo: | We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional \( s \)-gradient constraint, \( 0 < s < 1 \), associated to a general, possibly degenerate, linear fractional operator of the type, $$ L_s u = -D^s \cdot (A D^s u + bu) + d \cdot D^s u + cu, $$ with integrable data, in the space \( \Lambda^{s,p}_0(\Omega) \), which is the completion of the set of smooth functions with compact support in a bounded domain \( \Omega \) for the \( L_p \)-norm of the distributional Riesz fractional gradient \( D^s \) in \( \mathbb{R}^d \) (when \( s = 1 \), \( D_1 = D \) is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of \( L^\infty(\mathbb{R}^d) \) and are associated to the variational inequalities of the corresponding transport potentials under the constraint \( |D^s u| \leq g \). Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator \( L_s u \). For this purpose, we also develop some relevant properties of the spaces \( \Lambda^{s,p}_0(\Omega) \), including the limit case \( p = \infty \) and the continuous embeddings \( \Lambda^{s,q}_0(\Omega) \subset \Lambda^{s,p}_0(\Omega) \), for \( 1 \leq p \leq q \leq \infty \). We also show the localisation of the nonlocal problems \( (0 < s < 1) \), to the local limit problem with classical gradient constraint when \( s \rightarrow 1 \), for which most results are also new for a general, possibly degenerate, partial differential operator \( L_1 u \) only with integrable coeficients and bounded gradient constraint. |
|---|---|
| Autores principais: | Azevedo, Assis |
| Outros Autores: | Rodrigues, José Francisco; Santos, Lisa |
| Assunto: | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| Ano: | 2024 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| _version_ | 1866875555705520128 |
|---|---|
| author | Azevedo, Assis |
| author2 | Rodrigues, José Francisco Santos, Lisa |
| author2_role | author author |
| author_facet | Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
| author_role | author |
| contributor_name_str_mv | Universidade do Minho |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Azevedo, Assis\"},{\"Person.name\":\"Rodrigues, José Francisco\"},{\"Person.name\":\"Santos, Lisa\"}] |
| datacite.contributors.contributor.contributorName.fl_str_mv | Universidade do Minho |
| datacite.creators.creator.creatorName.fl_str_mv | Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
| datacite.date.Accepted.fl_str_mv | 2024-01-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2024-01-29T08:10:56Z |
| datacite.date.embargoed.fl_str_mv | 2024-01-29T08:10:56Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| datacite.titles.title.fl_str_mv | Nonlocal Lagrange multipliers and transport densities |
| dc.contributor.none.fl_str_mv | Universidade do Minho |
| dc.creator.none.fl_str_mv | Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
| dc.date.Accepted.fl_str_mv | 2024-01-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2024-01-29T08:10:56Z |
| dc.date.embargoed.fl_str_mv | 2024-01-29T08:10:56Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | https://hdl.handle.net/1822/88311 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | World Scientific |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| dc.title.fl_str_mv | Nonlocal Lagrange multipliers and transport densities |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional \( s \)-gradient constraint, \( 0 < s < 1 \), associated to a general, possibly degenerate, linear fractional operator of the type, $$ L_s u = -D^s \cdot (A D^s u + bu) + d \cdot D^s u + cu, $$ with integrable data, in the space \( \Lambda^{s,p}_0(\Omega) \), which is the completion of the set of smooth functions with compact support in a bounded domain \( \Omega \) for the \( L_p \)-norm of the distributional Riesz fractional gradient \( D^s \) in \( \mathbb{R}^d \) (when \( s = 1 \), \( D_1 = D \) is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of \( L^\infty(\mathbb{R}^d) \) and are associated to the variational inequalities of the corresponding transport potentials under the constraint \( |D^s u| \leq g \). Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator \( L_s u \). For this purpose, we also develop some relevant properties of the spaces \( \Lambda^{s,p}_0(\Omega) \), including the limit case \( p = \infty \) and the continuous embeddings \( \Lambda^{s,q}_0(\Omega) \subset \Lambda^{s,p}_0(\Omega) \), for \( 1 \leq p \leq q \leq \infty \). We also show the localisation of the nonlocal problems \( (0 < s < 1) \), to the local limit problem with classical gradient constraint when \( s \rightarrow 1 \), for which most results are also new for a general, possibly degenerate, partial differential operator \( L_1 u \) only with integrable coeficients and bounded gradient constraint. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| fulltext.url.fl_str_mv | https://repositorium.uminho.pt/bitstreams/61e09e15-ba5f-4cb9-9c4b-ea27c03f6f3b/download |
| id | rum_63dbabbdf499e859b7fb8ce58bbc2aef |
| identifier.url.fl_str_mv | https://hdl.handle.net/1822/88311 |
| instacron_str | repositorium |
| institution | Universidade do Minho |
| instname_str | Universidade do Minho |
| language | eng |
| network_acronym_str | rum |
| network_name_str | RepositóriUM - Universidade do Minho |
| oai_identifier_str | oai:repositorium.uminho.pt:1822/88311 |
| organization_str_mv | urn:organizationAcronym:repositorium |
| person_str_mv | Azevedo, Assis Rodrigues, José Francisco Santos, Lisa |
| publishDate | 2024 |
| publisher.none.fl_str_mv | World Scientific |
| reponame_str | RepositóriUM - Universidade do Minho |
| repository_id_str | urn:repositoryAcronym:rum |
| service_str_mv | urn:repositoryAcronym:rum |
| spelling | engWorld ScientificporWe prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional \( s \)-gradient constraint, \( 0 < s < 1 \), associated to a general, possibly degenerate, linear fractional operator of the type, $$ L_s u = -D^s \cdot (A D^s u + bu) + d \cdot D^s u + cu, $$ with integrable data, in the space \( \Lambda^{s,p}_0(\Omega) \), which is the completion of the set of smooth functions with compact support in a bounded domain \( \Omega \) for the \( L_p \)-norm of the distributional Riesz fractional gradient \( D^s \) in \( \mathbb{R}^d \) (when \( s = 1 \), \( D_1 = D \) is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of \( L^\infty(\mathbb{R}^d) \) and are associated to the variational inequalities of the corresponding transport potentials under the constraint \( |D^s u| \leq g \). Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator \( L_s u \). For this purpose, we also develop some relevant properties of the spaces \( \Lambda^{s,p}_0(\Omega) \), including the limit case \( p = \infty \) and the continuous embeddings \( \Lambda^{s,q}_0(\Omega) \subset \Lambda^{s,p}_0(\Omega) \), for \( 1 \leq p \leq q \leq \infty \). We also show the localisation of the nonlocal problems \( (0 < s < 1) \), to the local limit problem with classical gradient constraint when \( s \rightarrow 1 \), for which most results are also new for a general, possibly degenerate, partial differential operator \( L_1 u \) only with integrable coeficients and bounded gradient constraint.application/pdfporNonlocal Lagrange multipliers and transport densitiesAzevedo, AssisRodrigues, José FranciscoSantos, LisaHostingInstitutionOrganizationalUniversidade do Minhoe-mailmailto:repositorium@usdb.uminho.ptrepositorium@usdb.uminho.ptISSNIsPartOf1664-3607DOIIsPartOf10.1142/S16643607235001452024-01-29T08:10:56Z20242024-01-01T00:00:00ZHandlehttps://hdl.handle.net/1822/88311http://purl.org/coar/access_right/c_abf2open accessFractional gradientNonlocal variational inequalitiesLagrange multipliers631990 bytesliteraturehttp://purl.org/coar/resource_type/c_6501journal articlehttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://repositorium.uminho.pt/bitstreams/61e09e15-ba5f-4cb9-9c4b-ea27c03f6f3b/download |
| spellingShingle | Nonlocal Lagrange multipliers and transport densities Azevedo, Assis Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| status | SINGLETON |
| subject.fl_str_mv | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| title | Nonlocal Lagrange multipliers and transport densities |
| title_full | Nonlocal Lagrange multipliers and transport densities |
| title_fullStr | Nonlocal Lagrange multipliers and transport densities |
| title_full_unstemmed | Nonlocal Lagrange multipliers and transport densities |
| title_short | Nonlocal Lagrange multipliers and transport densities |
| title_sort | Nonlocal Lagrange multipliers and transport densities |
| topic | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| topic_facet | Fractional gradient Nonlocal variational inequalities Lagrange multipliers |
| url | https://hdl.handle.net/1822/88311 |
| visible | 1 |